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We derive the evolution equation of the average uncertainty energy forperiodic/homogeneous incompressible Navier-Stokes turbulence and show thatuncertainty is increased by strain rate compression and decreased by strainrate stretching. We use three different direct numerical simulations (DNS) ofnon-decaying periodic turbulence and identify a similarity regime where (a) theproduction and dissipation rates of uncertainty grow together in time, (b) theparts of the uncertainty production rate accountable to average strain rateproperties on the one hand and fluctuating strain rate properties on the otheralso grow together in time, (c) the average uncertainty energies along thethree different strain rate principal axes remain constant as a ratio of thetotal average uncertainty energy, (d) the uncertainty energy spectrum'sevolution is self-similar if normalised by the uncertainty's averageuncertainty energy and characteristic length and (e) the uncertainty productionrate is extremely intermittent and skewed towards extreme compression eventseven though the most likely uncertainty production rate is zero. Properties(a), (b) and (c) imply that the average uncertainty energy grows exponentiallyin this similarity time range. The Lyapunov exponent depends on both theKolmogorov time scale and the smallest Eulerian time scale, indicating adependence on random large-scale sweeping of dissipative eddies. In the two DNScases of statistically stationary turbulence, this exponential growth isfollowed by an exponential of exponential growth, which is in turn followed bya linear growth in the one DNS case where the Navier-Stokes forcing alsoproduces uncertainty.

The production of uncertainty in three-dimensional Navier-Stokes turbulence